2 Instructor : Larry Guth Transcribed by Ruoxuan

2 Instructor : Larry Guth Transcribed

2 Instructor : Larry Guth Transcribed

Instructor: Larry Guth Transcribed by Jonathan Tidor

One hint that Theorem 0.1 might be more tractable than Theorem 0.2 is that the maximum value of p is the critical exponent in multilinear restriction. Our proof of Theorem 0.1 indeed uses multilinear restriction as an input. This lecture is divided into 3 parts. First are the “multiscale tools”; a statement of some of the properties of the decoupling problem that makes arguments at different sc...

2 Instructor : Larry Guth Transcribed By

Question 1.2. Suppose g(k) is the least number of powers to make Theorem 1.1 hold. How big is g(k)? To provide some intuition, take n = 2−1. In its decomposition any ai can’t be greater than 1, since 2 k > 2 − 1. Therefore, ai = 1 for all i and g(k) ≥ s = 2 − 1. The real value of g(k) is only a little worse than that. Thus, the number g(k) grows at least exponentially with respect to k. However...

Instructor : Larry Guth Transcribed

Today we will talk about some applications of decoupling. The first application is Strichartz inequality on tori. Let T d = R/Z. For a lattice Λ ⊂ R, let TΛ = R/Λ. For instance, we may take the rectangle [0, λ1]× ...× [0, λd] and identify opposite edges. Our goal is to study the initial value problem for Schrodinger’s equation. That is, let u(x, t) : R×R→ C (or T ×R→ C or TΛ×R→ C), where x ∈ R,...